1. 1 Reflections Trondheim Sept 03

2. 2 / To be (an effective) teacher is to be ‘in question’, to be seeking to notice opportunities to act freshly and effectively – about mathematics – about learners experience – about self  Habits, ideals, actions, practices, …

3. 3 / ‘Being Mathematical’, with and in front of learners – means working on my ‘mathematical being’ – means explicitly using and developing natural powers with learners – means explicitly recognising mathematical themes, connections, …

4. 4 Children’s Mathematical Powers / Imagining & Expressing / Specialising & Generalising / Conjecturing & Convincing / Organising & Characterising / …

5. 5 Mathematical Themes / Doing & Undoing / Invariance in the Midst of Change / Freedom & Constraint / Extending & Contracting Meaning /…/…

6. 6 / (we) want learners (and teachers) to have positive self-affirming and esteem-enhancing yet challenging mathematical experiences / (we) want teachers to develop ways of working which offer these opportunities to learners

7. 7 / Professional development and personal development are inseparable / K subject : maths as living disciplined way of making sense of our worlds / K ped.content : didactics of specific topics / K ped : classroom and people management For me:

8. 8 / Maths Ed’n literature provides distinctions which may be helpful in – sensitising me to notice situations as phenomena – enabling me to discern detail – informing my choice of action in order to act freshly rather than out of habit For me:

9. 9 Example / Responses to suggestion: – “oh yes, I (we) do that already” – “there isn’t time, resource … to do that” – “hmm, what could I do differently to make that happen?” – Seen as examples of assimilating & accommodating

10. 10 / Reflection on experience does not guarantee sense-making, nor contact with didactics – One thing we do not seem to learn from experience is that we do not learn from experience alone / In years 1 & 2, need to build – Reflective practices – Mathematical practices / Pedagogic Issues & Didactic Distinctions – Ped: Gender issues, generating productive discussion – Did: Concept images, van Hiele, example-spaces, enactive- iconic-symbolic worlds, …

11. 11 DidacticsMathematics Pedagogy Perceived consistency between mathematics as encountered and didactics as espoused Topics, Themes, Powers Form of assessment determines student focus Course Structures Didactical Mathematics Topic particularities Tasks (structure, presentation) Activity (mediated actions) Interaction (mediated actions) Bringing to closure Gender issues, Discussion, … Fantastic window of opportunity to create something original and teacher-practice-oriented

12. 12 Didactical Mathematics / Core structure: significant didactical constructs (distinctions) and classroom tactics (practices) / Mathematical Topics used to illustrate these / Didactical constructs and tactics used to develop mathematical topics / Assessment is didactics oriented, not mathematical proficiency oriented

13. 13 Example of Possibility / Concept-Image – Various concepts just within and on the edge of current confidence / Example-Spaces – Extending personal spaces / Dimensions-of-possible-variation – As device to explore concepts (familiar, and new) – As device for opening up tasks for exploration – As device for extending personal spaces

14. 14 / Draw a quadrilateral / And another quadrilateral / Draw a quadrilateral with one pair of opposite sides perpendicular / Draw a quadrilateral with two pairs of opposite sides perpendicular

15. 15 Origins of Mathematics / Cultural inheritance (problems encountered) / Recognising mathematisable phenomena – Material-world situations – Imagery-world situations – Symbolic-world situations

16. 16 Mathematics Education as a Scientific Discipline / Particularising generalities / Generalisiang particularities (creating phenomena) / Problematising habits and assumptions / Locating and accessing didactic and pedagogic strategies and tactics